Interpretation of a ROR value

From the perspective of someone who has borrowed money, the interest rate is applied to the unpaid balance so that the total loan amount and interest are paid in full exactly with the last loan payment. From the perspective of a lender of money, there is an unrecovered balance at each time period. The interest rate is the return on this unrecovered balance so that the total amount lent and the interest are recovered exactly with the last receipt. Rate of return describes both of these perspectives.

The rate of return is expressed as a percent per period, for example, i = 10% per year. It is stated as a positive percentage; the fact that interest paid on a loan is actually a negative rate of return from the borrower’s perspective is not considered. The numerical value of i can range from -100% to infinity, that is, -100% ≤ i < ∞. In terms of an investment, a return of      i = - 100% means the entire amount is lost.

The definition above does not state that the rate of return is on the initial amount of the investment; rather it is on the unrecovered balance, which changes each time period. Example (6-1) illustrates this difference.

To get started in a new telecommuting position with AB Hammond Engineers, Jane took out a $1000 loan at i = 10% per year for 4 years to buy home office equipment. From the lender’s perspective, the investment in this young engineer is expected to produce an equivalent net cash flow of $315.47 for each of 4 years.

This represents a 10% per year rate of return on the unrecovered balance. Compute the amount of the unrecovered investment for each of the 4 years using

1-      The rate of return on the unrecovered balance (the correct basis)

2-      The return on the initial $1000 investment.

3-      Explain why all of the initial $1000 amount is not recovered by the final payment in part (2).

1-      Table (6–1) shows the unrecovered balance at the end of each year in column 6 using the 10% rate on the unrecovered balance at the beginning of the year. After 4 years the total $1000 is recovered, and the balance in column 6 is exactly zero.

Table (6-1): Unrecovered Balances Using a Rate of Return of 10% on the Unrecovered Balance

2-      Table 7–2 shows the unrecovered balance if the 10% return is always figured on the initial $1000. Column 6 in year 4 shows a remaining unrecovered amount of $138.12, because only $861.88 is recovered in the 4 years (column 5).

3-      As shown in column 3, a total of $400 in interest must be earned if the 10% return each year is based on the initial amount of $1000. However, only $261.88 in interest must be earned if a 10% return on the unrecovered balance is used. There is more of the annual cash flow available to reduce the remaining loan when the rate is applied to the unrecovered balance as in part (1) and Table (6–1) . Figure (6–1) illustrates the correct interpretation of rate of return in Table (6–1).

Figure (6-1): Plot of unrecovered balances and 10% per year rate of return on a $1000 amount, Table (6-1)

Each year the $315.47 receipt represents 10% interest on the unrecovered balance in column 2 plus the recovered amount in column 5.

Because rate of return is the interest rate on the unrecovered balance, the computations in Table (6–1) for part (1) present a correct interpretation of a 10% rate of return. Clearly, an interest rate applied only to the principal represents a higher rate than is stated. In practice, a so-called add-on interest rate is frequently based on principal only, as in part (2). This is sometimes referred to as the installment financing problem.

Installment financing can be discovered in many forms in everyday finances. One popular example is a “no-interest program” offered by retail stores on the sale of major appliances, audio and video equipment, furniture, and other consumer items. Many variations are possible, but in most cases, if the purchase is not paid for in full by the time the promotion is over, usually 6 months to 1 year later, finance charges are assessed from the original date of purchase. Further, the program’s fine print may stipulate that the purchaser use a credit card issued by the retail company, which often has a higher interest rate than that of a regular credit card, for example, 24% per year compared to 15% per year. In all these types of programs, the one common theme is more interest paid over time by the consumer. Usually, the correct definition of i as interest on the unpaid balance does not apply directly; i has often been manipulated to the financial disadvantage of the purchaser.